Summary:
This class introduces the concept of the Thermodynamic Limit to explain how some physical systems are treated statistically. An analogy of particles colliding with a wall is used, where pressure is defined as the total force per unit area. By considering an infinite area, pressure in a container is calculated based on the momentum exerted by the molecules on its walls.

Learning Objectives:
By the end of this class, the student will be able to:

1. Explain how the thermodynamic limit applies to the definition of pressure in terms of total force and area
2. Understand how the thermodynamic limit applies in statistical physics and the kinetic theory of gases.
3. Understand the difference between intensive and extensive variables.
4. Understand the basic ideas behind the different approaches to studying thermodynamics.

TABLE OF CONTENTS:
INTRODUCING PRESSURE IN THE THERMODYNAMIC LIMIT
THE THERMODYNAMIC LIMIT
EXTENSIVE AND INTENSIVE VARIABLES
APPROACHES TO THERMODYNAMICS

Introducing Pressure in the Thermodynamic Limit

The concept of the Thermodynamic Limit helps understand why certain physical systems can be treated using statistical considerations. This is due to the large number of particles that compose them. A simple way to demonstrate this is through an analogy. Imagine you have a particle cannon that shoots them at a certain speed against a wall; since the particles have mass, upon colliding with the wall, they transfer some of their momentum, thus exerting some impulse.

Thus, knowing the speed and mass, the force exerted by each particle can be calculated. Now imagine it’s not a cannon, but a uniformly concentrated shower of countless particles hitting a region of the ground, which can be as large as we want. What is the consequence of this?

1. The average force exerted increases as we consider a larger area. This makes sense because the larger the area, the more particles are received.
2. Although the force exerted by each particle fluctuates, this is “smoothed out” and tends to an average value. In fact, the fluctuations can be large, but if we increase the area, the total force will be so enormous that such fluctuation will be insignificant.

Since the total force exerted is proportional to the area, it makes sense to establish the following definition:

Definition

The Pressure P generated by a total force \vec{F} applied over an area {A} is defined as the limit \color{blue}{\displaystyle P = \lim_{A\to\infty} \frac{\vec{F}\cdot \hat{n}}{A}} Where \hat{n} is the vector normal to the surface. This is what is often written concisely as \displaystyle P = \frac{F}{A}

The pressure introduced in our analogy does not change as the area increases; on the contrary, pressure fluctuations tend to disappear. In fact, fluctuations can be ignored if we take the limit where the area tends to infinity.

The Thermodynamic Limit

If we consider the moving molecules inside a container, each time they collide with the boundary, they exert a certain impulse on it. The collective effect of all these impulses is what we interpret as pressure: a force per unit area extended over the entire surface. If the container were very small, perhaps we would need to worry about force fluctuations; however, in most cases, the number of particles is so large that fluctuations can be ignored. The pressure of a gas under these conditions is considered completely uniform. This description we just made is what is understood as “being in the thermodynamic limit.”

Extensive and Intensive Variables

Suppose a container with volume V has a gas at temperature T, pressure P, and its total kinetic energy is U. Now imagine we place a barrier inside the container, dividing the gas into two equal halves. Then the volume of each half V^* will be

\displaystyle V^* = \frac{V}{2}

The total kinetic energy of each half U^* will also be halved

\displaystyle U^* = \frac{U}{2}

However, quantities like temperature and pressure will remain the same in both halves

P^* = P

T^* = T

From this emerges a distinction between the magnitudes involved in thermodynamics. We refer to extensive variables when the magnitudes these represent scale with the size of the system, like volume or energy, and intensive variables as the magnitudes that do not scale with the size of the system, like pressure and temperature.

Approaches to Thermodynamics

Historically, thermodynamics has developed through different stages that have given us several approaches.

• Classical Thermodynamics deals with macroscopic properties such as pressure, temperature, and volume, without concerning itself with the microscopic aspects of matter. It deals with systems large enough to ignore the pre-thermodynamic limit fluctuations and disregards the atomic structure of matter.
• Kinetic Theory of Gases seeks to determine the properties of gases by considering the probability distributions associated with the movement of their molecules. Its beginning was controversial because, when it was developed, there were still doubts about the existence of atoms and molecules, which was demonstrated at the end of the 19th century.
• The discovery of atoms led to the development of Statistical Mechanics. Instead of starting with the description of macroscopic properties, as is done in thermodynamics, its approach starts by attempting to describe the states of individual microscopic systems and then, using statistical methods, it tries to infer the macroscopic properties of the system. This approach has benefited from the development of quantum mechanics because it allows the description of quantum micro-systems. Thus, what is described in thermodynamics is obtained as a limiting process of statistical mechanics in the thermodynamic limit.